From the Fundamental Theorem of Algebra to Astrophysics: a “harmonious” Path

The fundamental theorem of algebra (FTA) tells us that every complex polynomial of degree n has precisely n complex roots. The first published proofs (J. d’Alembert in 1746 and C. F. Gauss 1799) of this conjecture from the seventeenth century had flaws, though Gauss’s proof was generally accepted as correct at the time. Gauss later published three correct proofs of the FTA (two in 1816 and the last presented in 1849). It has subsequently been proved in a multitude of ways, using techniques from analysis, topology, and algebra; see [Bur 07], [FR 97], [Re 91], [KP 02], and the references therein for discussions of the history of FTA and various proofs. In the 1990s, T. Sheil-Small and A. Wilmshurst proposed to extend FTA to a larger class of polynomials, namely, harmonic polynomials. (A complex polynomial h(x, y) is called harmonic if it satisfies the Laplace equation 4h = 0, where 4 := ∂/dx + ∂/dy.) A simple complex-linear change of variables z = x + iy, z = x − iy allows us to write any complex valued harmonic polynomial of two variables in the complex form